Kepler’s laws

Ancient Astronomy Ancient civilizations observed the skies Many built structures to mark astronomical events for religious and practical reasons: Seasons, day light gain and loss etc Figure 2-1. Caption: Stonehenge. This remarkable site in the south of England was probably constructed as a primitive calendar or almanac. The inset shows sunrise at Stonehenge at the summer solstice. As seen from the center of the stone circle, the Sun rose directly over the “heel stone” on the longest day of the year. (English Heritage) Summer solstice sunrise at Stonehenge (2800B.C. till 1100B.C)

Big Horn Medicine in Wyoming Spokes of the Big Horn Medicine Wheel are aligned with rising and setting of Sun and other stars Features that align with rising and setting of the sun and other stars. Figure 2-2a. Caption: Observatories in the Americas. (a) The Big Horn Medicine Wheel in Wyoming, built by the Plains Indians, has spokes and other features that roughly align with risings and settings of the Sun and other stars.

Caracol temple in Mexico This temple at Caracol, in Mexico, has many windows that are aligned with astronomical events: Keeping track of the seasons Figure 2-2b. Caption: Observatories in the Americas. (b) The Caracol temple in Mexico, built by the Mayan civilization, has some windows that seem to align with astronomical events, suggesting that at least part of Caracol’s function may have kept track of the seasons and the heavens.

The Geocentric Universe Aristotle (384-322 B.C.) Ancient astronomers observed: Sun Moon Stars Five planets: Mercury, Venus, Mars, Jupiter, Saturn Figure 2-3. Caption: Turkish Astronomers at Work. During the Dark Ages, much scientific information was preserved and new discoveries were made by astronomers in the Islamic world, as depicted in this illustration from a 16th-century manuscript. (The Granger Collection)

Sun, Moon, and stars all have simple movements in the sky Planets motion are more complicated: Move with respect to fixed stars Change in brightness Change speed Undergo retrograde motion Figure 2-4. Caption: Planetary Motion. Most of the time, planets move from west to east relative to the background stars. Occasionally—roughly once per year—however, they change direction and temporarily undergo retrograde motion (east to west) before looping back. The main illustration shows an actual retrograde loop in the motion of the planet Mars. The inset depicts the movements of several planets over the course of several years, as reproduced on the inside dome of a planetarium. The motion of the planets relative to the stars (represented as unmoving points) produces continuous streaks on the planetarium “sky.” (Boston Museum of Science)

Inferior planets: Mercury, Venus Superior planets: Mars, Jupiter, Saturn Now know: Inferior planets have orbits closer to Sun than Earth’s Superior planets’ orbits are farther away Figure 2-5. Caption: Inferior and Superior Orbits. Diagram of Earth’s orbit and two other possible planetary orbits. An “inferior” orbit lies between Earth’s orbit and the Sun. Mercury and Venus move in such orbits. A “superior” orbit (such as the orbit of Mars, Jupiter, or Saturn) lies outside that of Earth. The points noted on the orbits indicate times when a planet appears to come close to the Sun (conjunction) or is diametrically opposite the Sun on the celestial sphere (opposition); they are discussed further in the text. The inset is a schematic representation of the orbit of an inferior planet relative to the Sun, as seen from Earth.

The Geocentric Universe Earliest models had Earth at center of solar system Needed lots of complications to accurately track planetary motions Figure 2-7. Caption: Ptolemaic Model. The basic features, drawn roughly to scale, of Ptolemy’s geocentric model of the inner solar system, a model that enjoyed widespread popularity prior to the Renaissance. Only the five planets visible to the naked eye and hence known to the ancients—Mercury, Venus, Mars, Jupiter, and Saturn—are shown. The planets’ deferents were considered to move on spheres lying within the celestial sphere that held the stars. The celestial sphere carried all interior spheres around with it, but the planetary (and solar) spheres had additional motions of their own, causing the Sun and planets to move relative to the stars. To avoid confusion, partial paths (dashed) of only two planets—Venus and Jupiter—are drawn here. Cosmic lecture launcher:3.15 Ptolemy’s model

The scientific method New observations (retrograde motion, phases of Venus) lead to new model: Heliocentric universe Heliocentric universe allows for new predictions: Phases and retrograde motion of other planets

The Foundations of the Copernican Revolution (1473-1543) Earth is not at the center of everything. Center of earth is the center of moon’s orbit. All planets revolve around the Sun. The stars are very much farther away than the Sun. The apparent movement of the stars around the Earth is due to the Earth’s rotation. The apparent movement of the Sun around the Earth is due to the Earth’s rotation. Retrograde motion of planets is due to Earth’s motion around the Sun.

Kepler’s Laws of Planetary Motion Kepler’s laws were derived using observations made by Tycho Brahe(1546-1601) Figure 2-15. Caption: Tycho Brahe. The astronomer in his observatory Uraniborg, on the island of Hveen in Denmark. Brahe’s observations of the positions of stars and planets on the sky were the most accurate and complete set of naked-eye measurements ever made. (Royal Ontario Museum) Kepler: 1571-1630 AC

Kepler’s 1’st law First Law: Planetary orbits are ellipses (not circular), Sun at one focus Cosmic lecture launcher: Drawing an ellipse. Figure 2-16. Caption: Ellipse. An ellipse can be drawn with the aid of a string, a pencil, and two thumbtacks. The wider the separation of the foci, the more elongated, or eccentric, is the ellipse. In the special case where the two foci are at the same place, the curve drawn is a circle.

Some Properties of Planetary Orbits Semimajor axis and eccentricity of orbit completely describe it Perihelion: closest approach to Sun Aphelion: farthest distance from Sun e=c/a Circle: c=0, e=0 a c x

Question Kepler’s 1st law of planetary orbits states that 1) planets orbit the Sun. 2) orbits are noncircular. 3) orbits are elliptical in shape. 4) all of the above Kepler’s 1st law of planetary orbits states that Answer: 4

Question Kepler’s 1st law of planetary orbits states that 1) planets orbit the Sun. 2) orbits are noncircular. 3) orbits are elliptical in shape. 4) all of the above Kepler’s 1st law of planetary orbits states that Kepler’s Laws apply to all orbiting objects. The Moon orbits Earth in an ellipse, and the Space Shuttle orbits Earth in an ellipse, too.

Kepler’s 2’nd law Second Law: Imaginary line connecting Sun and planet sweeps out equal areas in equal times Areas: Area A= Area B = Area C 1 2 3 4 5 6 Times: t2-t1=t4-t3=t6-t1 Figure 2-17. Caption: Kepler’s Second Law. A line joining a planet to the Sun sweeps out equal areas in equal intervals of time. The three shaded areas A, B, and Care equal. Any object traveling along the elliptical path would take the same amount of time to cover the distance indicated by the three red arrows. Therefore, planets move faster when closer to the Sun. Distance traveled: d12 > d34>d56

Question Earth orbits slower in January. Earth orbits faster in January. Earth’s orbital speed doesn’t change. Earth is closer to the Sun in January. From this fact, Kepler’s 2nd law tells us Answer: 2

Question Earth orbits slower in January. Earth orbits faster in January. Earth’s orbital speed doesn’t change. Earth is closer to the Sun in January. From this fact, Kepler’s 2nd law tells us Kepler’s 2nd law means that a planet moves faster when closer to the star. Faster Slower

Kepler’s 3’rd law Third Law: Square of period of planet’s orbital motion is proportional to cube of semimajor axis: Orbits must be given in AU and periods in Earth years!

Question speed. period. shape. velocity. Kepler’s 3rd law relates a planet’s distance from the Sun and its orbital Answer: 2

Question 13 speed. period. shape. velocity. Kepler’s 3rd law relates a planet’s distance from the Sun and its orbital Venus’ Period = 225 days Venus’ axis = 0.7 AU Kepler’s 3rd law P2 = a3 means more distant planets orbit more slowly. Earth’s Period = 365 days Earth’s axis = 1.0 AU

The Dimensions of the Solar System Now measured using radar: Ratio of mean radius of Venus’s orbit to that of Earth very well known Figure 2-19. Caption: Astronomical Unit. Simplified geometry of the orbits of Earth and Venus as they move around the Sun. The wavy blue lines represent the paths along which radar signals are transmitted toward Venus and received back at Earth at the particular moment (chosen for simplicity) when Venus is at its minimum distance from Earth. Because the radius of Earth’s orbit is 1 AU and that of Venus is about 0.7 AU, we know that this distance is 0.3 AU. Thus, radar measurements allow us to determine the astronomical unit in kilometers.